Simplify and expand the following expression: $ \dfrac{2}{5n - 25}- \dfrac{1}{5n + 15}- \dfrac{5n}{n^2 - 2n - 15} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{2}{5n - 25} = \dfrac{2}{5(n - 5)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{1}{5n + 15} = \dfrac{1}{5(n + 3)}$ We can factor the quadratic in the third term: $ \dfrac{5n}{n^2 - 2n - 15} = \dfrac{5n}{(n - 5)(n + 3)}$ Now we have: $ \dfrac{2}{5(n - 5)}- \dfrac{1}{5(n + 3)}- \dfrac{5n}{(n - 5)(n + 3)} $ The least common multiple of the denominators is: $ 25(n - 5)(n + 3)$ In order to get the first term over $25(n - 5)(n + 3)$ , multiply by $\dfrac{5(n + 3)}{5(n + 3)}$ $ \dfrac{2}{5(n - 5)} \times \dfrac{5(n + 3)}{5(n + 3)} = \dfrac{10(n + 3)}{25(n - 5)(n + 3)} $ In order to get the second term over $25(n - 5)(n + 3)$ , multiply by $\dfrac{5(n - 5)}{5(n - 5)}$ $ \dfrac{1}{5(n + 3)} \times \dfrac{5(n - 5)}{5(n - 5)} = \dfrac{5(n - 5)}{25(n - 5)(n + 3)} $ In order to get the third term over $25(n - 5)(n + 3)$ , multiply by $\dfrac{25}{25}$ $ \dfrac{5n}{(n - 5)(n + 3)} \times \dfrac{25}{25} = \dfrac{125n}{25(n - 5)(n + 3)} $ Now we have: $ \dfrac{10(n + 3)}{25(n - 5)(n + 3)} - \dfrac{5(n - 5)}{25(n - 5)(n + 3)} - \dfrac{125n}{25(n - 5)(n + 3)} $ $ = \dfrac{ 10(n + 3) - 5(n - 5) - 125n} {25(n - 5)(n + 3)} $ Expand: $ = \dfrac{10n + 30 - 5n + 25 - 125n}{25n^2 - 50n - 375} $ $ = \dfrac{-120n + 55}{25n^2 - 50n - 375}$ Simplify: $ = \dfrac{-24n + 11}{5n^2 - 10n - 75}$